Re: [Phys-l] Simulating disturbances in a stable planetary system.

[Ludwik Kowalski <kowalskil@mail.montclair.edu>]

On Jan 2, 2008, at 3:06 PM, John Denker wrote:

> On 01/02/2008 10:32 AM, chuck britton wrote:
>
> > How about the term 'equilibrium'?
> >
> > I think that it might more clearly relate to this discussion.
> >
> > Two body orbits remain in equilibrium when disturbed but are no
> > longer in the same 'state'.
>
> No, you don't want to go down that rabbit-hole.
>
> In mechanics, the terms equilibrium, stability, and damping
> are well defined.  The definitions are not likely to change
> any time soon.
>
> A pedagogical discussion of this, including some useful diagrams,
> can be found at
>   http://www.av8n.com/how/htm/equilib.html
>
> > A reclining cone remains in equilibrium when disturbed but not in the
> > same state.
>
> OK.
>
> > A standing cone is in equilibrium so dramatically that it is often
> > referred to as Stable Equilibrium.
>
> "Drama" is not part of the definition.
>
> > It is of course more rightly seen as a Meta-Stable equilibrium when
> > the disturbance is comparable to the 'depth of the equilibrium well'.
>
> The difference between stability and metastability is usually
> only a matter of degree.  Absolute, unconditional stability is
> rare and hard to achieve.
>
> > Feynmann was certainly correct when he would scorn the
> > 	'Give it a Name and we'll Understand it'
>
> Yeah, but the funny thing is that students still want names.
> Students overestimate the importance of terminology by a
> factor of 100 or so.
>
> There *is* a cost to not knowing the terminology, as Feynman
> well knew
>    
> http://www.multitran.ru/c/m.exe? 
> a=DisplayParaSent&fname=Richard%20Feynman%5CChapter09
>
> I like to say that ideas are primary and terminology is
> secondary.  Terminology is useful to the extent that it
> helps us formulate and communicate the ideas.
>
>   In contrast, terminology for the sake of terminology
>   is a colossal waste of time.
>
>   Loosely speaking, terminology is like laboratory apparatus,
>   in the following sense:  One of my favorite John Reppy quotes
>   is:
>     "You should never build any more apparatus than you need,
>      or any less."

1) In the last two days i was playing with my own planetary simulation  
program (one planet m<<M). Its output is not as fancy as that in I.P.  
(commercial Interactive Physics software) but it allows me to create  
disturbances that I was not able to produce in I.P. Both programs  
produce identical results for unperturbed motions when initial  
conditions are the same. That was done to check that no serious errors  
were made in my code.

2) My program was used with different kinds of disturbances, such a  
change in position (suddenly moving a planet into a position in which  
the fields is either stronger or weaker), or a sudden change in  
velocity, or both, etc. I never observed the effect which, according to  
JohnD, would be called "positive stability" (see the first URL above).  
Disturbances change trajectories, for example, a circular orbit becomes  
elliptical. Or an elliptical orbit becomes longer. In the absence of  
disturbances each orbit is persistent/durable (periodic). A motion in a  
state of "negative stability" (usually called unstable) could also be  
produced. These were cases in which the escape velocity was exceeded by  
the planet. On that basis one can say that a two-body planetary system  
is never "positively stable." The potential energy does not have a  
minimum, when plotted against the distance between two objects. That is  
why positive stability does not happen.

3) By the way, consider an elliptical trajectory and plot the distance  
r versus time. You will see oscillations between r(min) and r(max). The  
cycles repeat themselves periodically. But changes in r near the r(min)  
are more rapid than changes in r near r(max); the d(t) is not a sin  
curve. The potential energy, U, is inversely proportional to d. Thus  
maxima in r(t) correspond to minima in U(t), and vice versa.  There is  
nothing profound in this. But I find it interesting.

_______________________________________________________
Ludwik Kowalski, a retired physicist
5 Horizon Road, apt. 2702, Fort Lee, NJ, 07024, USA
Also an amateur journalist at http://csam.montclair.edu/~kowalski/cf/